# CosFT

CosFT, or Cosinus Transform, refers to the integral transform with kernel $$K(x,y)=\sqrt{\frac{2}{\pi}} \cos(xy)$$;

for function $$f$$, the CosFT$$f$$ appears as $$g$$ defined with

$$\displaystyle g(x)=\,$$CosFT$$f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \cos(xy) \, f(y) \, \mathrm d y$$

## SinFT and CosFT

CosFT often appears together with SinFT;

the sine transform SinFT of function $$f$$ appears as $$g=\,$$SinFT$$f$$ with rofmula

$$\displaystyle g(x)=\sqrt{\frac{2}{\pi}} \int_0^\infty \sin(xy) \, f(y) \, \mathrm d y$$

It is assumed that function $$f$$ decays (or, at least, quickly oscillates) at infinity, in such a way that the integral converges.

Then, SinFT$$^2=\,$$CosFT$$^2=\hat 1$$, id est, the identity transform.

## Numerical implementation

CosFT can be implemented numerically through the CFT transform at the uniform grid at $$N\!+\!1$$ nodes; for array $$f$$, the SFT $$g$$ is defined with

$$g_m=\,$$CFT$$\displaystyle f_m=$$ $$\displaystyle \frac{1}{2}\, f_0+\frac{(-1)^m}{2}\, f_{N} + \sum_{n=1}^{N-1} \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n$$

The Numerical recipes in C (http://numerical.recipes) suggest the implementation through routines four1 and realft; however, for the serious applications, specification "float" should be replaced to something appropriate, for example, double, or complex double. [1][2][3][4]

At given number $$N$$ of nodes, the set of the nodes can be denoted with $$x_n$$ for $$n=0 .. N$$,

$$\displaystyle x_n=\sqrt{\frac{\pi}{N}}~ n$$

then, for $$f_n=f(x_n)$$, at large $$N\gg 1$$, the transform $$~g(x)=\frac{2}{\pi}\int_0^\infty f(y)\,\cos(x\,y)\,\mathrm d y~$$ is approximated with

$$\displaystyle g(x_m) \approx g_m = \sqrt{\frac{2}{N}}\, \left( \frac{f_0+(-1)^m f_N}{2} + \sum_{n=1}^{N-1} \, \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n\right)$$

## Eigenfunctions

Eigenfunctions $$F$$ of the CosFT appear as even Oscillator functions.

The simplest of them is just Gaussian;

$$F(x)=\exp(-x^2/2)$$

This property is used in the C++ test. The thick segmented line in the explicit plots at right show the discrete representation of the Gaussian at the discrete grid with $$N\!=\!4$$ and $$N\!=\!16$$.

The thin segmented line shows its SinFT transform, as it is approximated at this grid. The discrete representation and its CFT practically coincide; the deviation is smaller than the thicknesses of the lines.